Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds

Abstract

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto {\lambda }_1(G/K,g)\mathrm{diam}(G/K,g{)}^2$ is bounded among $G$-invariant metrics $g$ on $G/K$. Eldredge, Gordina, and Saloff-Coste conjectured in 2018 that this assertion holds when $K$ is trivial; the only particular cases known so far are when $G$ is abelian, $\mathrm{SU}(2)$, and $\mathrm{SO}(3)$. In this article we prove the existence of the mentioned upper bound for every compact homogeneous space $G/K$ having multiplicity-free isotropy representation.

Accepted for publication in Transformation Groups. arXiv admin note: text overlap with arXiv:2004.00350

References in corpus (2)

• The first eigenvalue of a homogeneous CROSS
• Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups